Problem 20 - Calculus: Early Transcendentals

Question

Find a power series representation for the function and determine the radius of convergence. \( f(x) = \frac {x^2 + x}{(1 - x)^3} \)

Step-by-Step Solution

Answer

The series representation is \( \sum_{n=0}^{\infty} [\binom{n+2}{2} x^{n+2} + \binom{n+2}{2} x^{n+1}] \) with radius of convergence \( |x| < 1 \).

Step 1: Identify Form for Series Expansion

Recognize that the denominator \( (1-x)^3 \) is similar to the binomial series expansion form \( \frac {1}{(1-x)^n} \). This suggests the known binomial series expansion: \[ \frac {1}{(1-x)^3} = \sum_{n=0}^{\infty} \binom{n+2}{2} x^n \] for \( |x| < 1 \).

Step 2: Break Function into Components

Split the given function into two parts as follows: \( f(x) = x^2 \cdot \frac{1}{(1-x)^3} + x \cdot \frac{1}{(1-x)^3} \). Each component can be individually expanded using the binomial series obtained in Step 1.

Step 3: Expand Each Component Separately

Expand the first component \( x^2 \cdot \frac{1}{(1-x)^3} \) using the binomial series: \[x^2 \cdot \frac{1}{(1-x)^3} = x^2 \cdot \sum_{n=0}^{\infty} \binom{n+2}{2} x^n = \sum_{n=0}^{\infty} \binom{n+2}{2} x^{n+2}\]Similarly, expand the second component \( x \cdot \frac{1}{(1-x)^3} \): \[ x \cdot \frac{1}{(1-x)^3} = x \cdot \sum_{n=0}^{\infty} \binom{n+2}{2} x^n = \sum_{n=0}^{\infty} \binom{n+2}{2} x^{n+1} \]

Step 4: Combine Expansions

Combine the expanded series from each component: \[f(x) = \sum_{n=0}^{\infty} \binom{n+2}{2} x^{n+2} + \sum_{n=0}^{\infty} \binom{n+2}{2} x^{n+1}\]Re-index these series to start from the same exponent if needed.

Step 5: Determine Radius of Convergence

Use the known radius of convergence for a geometric series: For \( \sum_{n=0}^{\infty} c_n x^n \), the radius of convergence \( R = 1\) as the original binomial expansion \( \sum_{n=0}^{\infty} \binom{n+2}{2} x^n \) is valid for \( |x| < 1 \).

Key Concepts

Radius of Convergence

The radius of convergence is a key concept when dealing with power series. It defines the interval, or the set of input values, within which the power series converges — essentially indicating where the sum of the series gives valid results.

To determine the radius of convergence, you can often use a method called the Ratio Test or an established rule for specific types of series.

• For geometric series, the radius of convergence is found directly from the condition that ensures convergence: for a series like \[ \sum_{n=0}^{\infty} c_n x^n \] if the series converges when \( |x| < R \), where \( R \) is the radius of convergence.
• In the binomial series expansion, as in this exercise, for \( \left| x \right| < 1 \), the binomial series is valid, thus giving a radius of convergence \( R = 1 \).

Understanding the radius of convergence helps us know the limits within which our power series faithfully recovers functions. Always identify this when working with power series.

Binomial Series

The binomial series is a specific type of series expansion that extends the binomial theorem to powers of any real or complex number.

• Specifically, it is used when you encounter expressions like \( \frac{1}{(1-x)^n} \).
• The general form of a binomial series is represented as \[ \frac{1}{(1-x)^n} = \sum_{k=0}^{\infty} \binom{n+k-1}{k} x^k \]
• This generalization helps in simplifying expressions when \( |x| < 1 \).

In this exercise, we use the binomial series to expand \( \frac{1}{(1-x)^3} \), enabling power series representation that fits well with our problem. By breaking down the function and applying this pattern, it offers a clear path to simplify and address complex denominators.

Series Expansion

Series expansion involves expressing a function as an infinite sum of terms. By converting a function into a power series, you can simplify calculations and gain insights into its behavior over a defined interval.

• Here, we express the original function \( f(x) = \frac {x^2 + x}{(1 - x)^3} \) in terms of a power series.
• It involves recognizing familiar patterns (e.g., binomial series) and applying them to each part of the function separately.
• This step-by-step expansion allows the simplification of each component before recombining the series to form a total power series.

The goal of series expansion is to rewrite familiar and complex expressions into simpler, more usable forms. This technique is powerful in calculus and beyond for approximating functions and solving differential equations.

Geometric Series

A geometric series is a simple but essential concept in mathematics that describes a series with a constant ratio between successive terms.

• Consider the series \( a + ar + ar^2 + ar^3 + \ldots \) with constant ratio \( r \).
• For a geometric series to converge, the absolute value of the ratio must be less than one, \( |r| < 1 \).
• The sum \( S \) of such a geometric series can be calculated using the formula \( S = \frac{a}{1-r} \).

While the function in the exercise doesn't directly represent a single geometric series, the convergence condition is crucial for determining its radius of convergence. Geometric series principles underpin much of the theoretical framework for series expansions, making them vital for understanding the nature of convergence and divergence in power series.